2 Flood Frequency Analysis Statistical Methods to evaluate probability exceeding a particular outcome - P (X >20,000 cfs) = 10%Used to determine return periods of rainfall or flowsUsed to determine specific frequency flows for floodplain mapping purposes (10, 25, 50, 100 yr)Used for datasets that have no obvious trendsUsed to statistically extend data sets
3 Basics of Probabilistic Prediction of Peak Events “What has happened and the frequency of events in a record are the best indicators of what can happen and its probability of happening in the future.”Requires a hydrological record at a station.The record is analyzed to estimate the probability of events of various sizes, as if they were independent of one another.Then it is extrapolated to larger, rarer floods.BUT most hydrologic records are short and non-stationary (i.e. conditions of climate and watershed condition change during the recording period). Magnitude of this problem varies, but needs to be checked in each case.
4 Return Period (Recurrence Interval) It is the average period in which a certain magnitude of a hydrological event will be once equalled or exceeded.the Weibull formulami is the rank of the ith event in a set of n eventsTr is the recurrence interval (yr). The long-term average intervalbetween events greater than Qi
5 Return period (Recurrence interval) cont. E.g. if the probability of exceeding 20 m3/sec in any year at a station is 0.01,It means that: there should be on average 10 events larger than 20 m3/sec in 1000 years, if the conditions affecting events at the site do not change.The average recurrence interval between events is 100 years, so we refer to such a discharge as “the 100-year event”.The events will not occur regularly every 100 years
6 How to compute the Return Period 1. Arrange the data in ascending or descending order of magnitude ignoring their chronological sequence.2. Each observation in the data has the same relative frequency of occurrence.3. For n observations, fr=1/n.4. For the min value of data we assign 100%.5. For the next smallest of the data we assign 100%- (1/n)*100%.6. Next, 100%- 2(1/n)*100%, …, etc. and end up of (1/n)*100% at the max value.Conversely, we can do the same form the max value to the min value (ascending).
7 Flood Frequency Curvea flood-frequency curve: It is a plot of the calculated values of Tr against Qi.
8 Plotting PositionIt is related to the the return period. It can be expressed in terms of relative frequency of occurrence or the probability that a hydrological event of a given magnitude will occur.The plotting position should satisfy some conditions (Gumbel, 1960):The probability that the magnitude of a hydrological variable, X, equals or exceeds an indicated value, a, once in a return period of Tr years is,2. The probability should be as close as possible to the observed frequency.3. Neither zero or one.
9 Methods to compute p and Tr Where,m = rank or order of X (m=1 for the largest value, 2 for the second largest value, …, n for the smallest value.n= sample size.
11 Choosing Theoretical Probability Distribution of an Event Note that this procedure involves fitting a theoretical probability distribution to an observed sample drawn from an imaginary (but not well-understood) populationThe “true” theoretical probability distribution of the events is not known, and we have no reason to believe it is simple or has only 1 or parameters.Plotting the data set on various types of graph paper with different scales, designed to represent various theoretical probability distributions as straight lines, yields graphs of different shapes, which when extrapolated beyond the limits of measurement predict a range of events.
12 Normal Probability Paper Normal Prob Paper converts the Normal CDF S curve into a straight line on a prob scale
18 Example 3 Year Rank Ordered cfs 1940 1 42,700 1925 2 31,100 1932 3 20,7001966419,3001969514,200198261988712,1001995810,3002000………….
19 Flood frequency plots of same record on different probability papers Log-extreme value paperGumbel Type IIILog-probability paper
20 How to Choose a Theoretical Probability Distribution ? One common approach to choice of flood frequency plotting paper is to choose one on which the observed data plot as a straight line that can be extrapolated to estimate rare, large events.A second is to choose one of the common ones, but this still leads to different predictions among analysts
21 Some Guidelines Take Mean and Variance (S.D.) of ranked data Take Skewness Cs of data (3rd moment about mean)If Cs near zero, assume normal dist’nIf Cs large, convert Y = Log x - (Mean and Var of Y)Take Skewness of Log data - Cs(Y)If Cs near zero, then fits LognormalIf Cs not zero, fit data to Log Pearson III
22 Siletz River Example 75 data points Original QY = Log QMean20,4524.2921Std Dev60890.129Skew0.7889Coef of Variation0.2980.03
23 Siletz River Example - Fit Normal and LogN Normal Distribution Q = Qm + z SQQ100 = (6089) = 34,620 cfsMean z (S.D.)Where z = std normal variate - tablesLog N Distribution Y = Ym + k SYY100 = (0.129) =k = freq factor and Q = 10Y = 39,100 cfs
24 Log Pearson Type III Y = Ym + k SY K is a function of Cs and Recurrence IntervalTable 3.4 lists values for pos and neg skewsFor Cs = -0.15, thus K = 2.15Y100 = (0.129) = 4.567Q = 10Y = 36,927 cfs for LP IIIPlot several points on Log Prob paper
25 LogN Prob Paper for CDFWhat is the prob that flow exceeds some given value yr valuePlot data with plotting position formula P = m/n+1 , m = rank, n = #Log N dist’n plots as straight line
26 LogN Plot of Siletz R.MeanStraight Line Fits Data Well
27 Siletz River Flow Data Various Fits of CDFs LP3 has curvature LN is straight line
28 Risk Assessment (Binomial Distribution) The probability of getting x successes followed by n-x failuresis the product of prob of n independent events: px (1-p)n-xThis represents only one possible outcome. The number of waysof choosing x successes out of n events is the binomial coeff. Theresulting distribution is the Binomial or B(n,p).
29 Risk and ReliabilityThe probability of at least one success in n years, where the probability of success in any year is 1/T,is called the RISK.Prob success = p = 1/T and Prob failure = 1-pRISK = 1-P(0)= 1 - Prob(no success in n years)= 1 - (1-p) n= 1 - (1 - 1/T) nReliability = (1 - 1/T) n
30 Risk ExampleWhat is the probability of at least one 50 yr flood in a 30 year advance period, where the probability of success in any year is P=1/T = 0.02RISK = 1 - (1 - 1/T)n = 1 - ( )30= 1 - (0.98)30 = or 46%If this is too large a risk, then increase design level to the 100 year where p = 0.01RISK = 1 - (0.99)30 = 0.26 or 26%